[Paper Review] Orthogonal polynomials in and on a quadratic surface of revolution

Sheehan Olver, Yuan Xu|arXiv (Cornell University)|Jun 25, 2019

Electromagnetic Scattering and Analysis8 references4 citations

TL;DR

This paper constructs explicit orthogonal polynomial bases for functions on and within quadratic surfaces of revolution—such as cones, hyperboloids, and paraboloids—generalizing spherical harmonics and ball-based orthogonal polynomials. Using fast transforms and cubature rules, it enables efficient high-accuracy function approximation and spectral methods on these non-standard domains.

ABSTRACT

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

Motivation & Objective

• To generalize spherical harmonics and classical orthogonal polynomials in the ball to quadratic surfaces of revolution in R^{d+1} for d ≥ 2.
• To develop explicit orthogonal polynomial bases for functions on the surface and inside the volume bounded by these surfaces.
• To enable high-accuracy approximation of smooth functions on such domains using spectral methods.
• To construct efficient cubature rules and fast algorithms for orthogonal expansions on these exotic geometries.
• To extend recent advances in fast transforms (e.g., Slevinsky's methods) to non-spherical, quadratic domains for practical computation.

Proposed method

• Define weight functions on quadratic surfaces of revolution and construct orthogonal polynomial bases via separation of variables and special function theory.
• Use tensor product bases in cylindrical coordinates (r, θ, t) to represent functions on cones and other surfaces, leveraging Jacobi and Chebyshev polynomials.
• Apply Slevinsky’s fast transform for orthogonal polynomials on the disk and sphere to re-expand functions efficiently with quasi-optimal complexity.
• Construct cubature rules based on product-type quadrature on tensor product grids of Gauss–Jacobi and Gauss–Chebyshev points.
• Use linear algebra to invert transformation matrices between tensor product bases and orthogonal polynomial bases, ensuring exact recovery of polynomial expansions.
• Leverage analyticity properties of functions to predict coefficient decay rates in orthogonal expansions, using singularities as predictors of convergence speed.

Experimental results

Research questions

• RQ1 How can orthogonal polynomials be explicitly constructed on the surface and interior of quadratic surfaces of revolution, such as cones and hyperboloids?
• RQ2 What is the structure of the orthogonal polynomial basis for a general weight function on these surfaces, and how does it generalize spherical harmonics?
• RQ3 Can fast algorithms for orthogonal expansions be adapted to non-spherical, quadratic domains using existing spectral transform techniques?
• RQ4 What is the rate of decay of expansion coefficients for analytic functions on these surfaces, and how does it relate to the location of singularities?
• RQ5 How can efficient cubature rules be constructed for integration over quadratic bodies of revolution using orthogonal polynomial bases?

Key findings

• Explicit orthogonal polynomial bases are constructed for functions on the surface and inside the volume bounded by cones, hyperboloids, and paraboloids in R^{d+1}.
• The orthogonal polynomials on the cone surface generalize spherical harmonics and are constructed using associated Legendre polynomials and Jacobi polynomials.
• For smooth functions on the cone, the coefficients in the orthogonal expansion decay exponentially, with the rate determined by the location of the nearest singularity.
• Entire functions in x, y, t on the cone exhibit super-exponential coefficient decay, indicating spectral convergence.
• The fast transform method based on Slevinsky’s algorithms achieves quasi-optimal complexity and enables efficient computation of orthogonal expansions.
• Cubature rules based on tensor product quadrature grids are derived and shown to be effective for integration and approximation on these domains.